Calculating beta for two moving light sources

Two monochromatic light sources approach each other head-on with equal
speeds, [itex]v=\beta c[/itex] relative to the laboratory. When they pass each other, the
frequency of the light each receives from the other is halved. Show that [itex]\beta =3-\sqrt{8}[/itex]

1. You should have available somewhere the formula for Doppler frequency modification for a given relative speed between an observer and a light source. For example, Halliday & Resnick, section 40.5. Doesn't look like you have the right formula (what is γ?) for that.

2. Use that formula to compute the sources' speed relative to each other.

3. Then compute how fast (including sign) each light source "sees" the lab whizzing by. In other words, consider inertal reference frames S1 and S2 as belonging to the two sources, respectively, in your formula (1) above.

from law of addition of velocities,you will have
u=2βc/(1+β^2) because in case of light only relative motion matters unlike sound.
when it is approaching then ,the formula you have written is right. but when it is going away then numerator and denominator interchanged.
so,
√[(1-b)/(1+b)]=(1/2)√[(1+b)/(1-b)] where b=2β/1+β^2.
this gives,
β=(√2-1)/(√2+1)=3-√8

from law of addition of velocities,you will have
u=2βc/(1+β^2) because in case of light only relative motion matters unlike sound.
when it is approaching then ,the formula you have written is right. but when it is going away then numerator and denominator interchanged.
so,
√[(1-b)/(1+b)]=(1/2)√[(1+b)/(1-b)] where b=2β/1+β^2.
this gives,
β=(√2-1)/(√2+1)=3-√8

I had worked out both these parts previously which shows i was along the right tracks. However, why do you let the value of u equal to be so that i can be substituted in?